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equations simultaneously, so its coordinates lie on both graphs. Such coordinates are precisely the points where the line and quadratic curve intersect."},{"text":"The turning points of the line and the curve.","correct":false,"explanation":"A turning point describes where a curve changes direction, and a straight line has no turning point. The solutions are the points where the two graphs intersect."},{"text":"The points of intersection between the curve and the axes.","correct":false,"explanation":"Intercepts with the axes solve the quadratic together with $x=0$ or $y=0$, not with the given linear equation. The required solutions are where the line meets the curve."},{"text":"The midpoints between the line and the curve.","correct":false,"explanation":"A midpoint is calculated from two known coordinates and need not lie on either graph. Simultaneous solutions are the points that lie on both the line and the curve."}]},{"id":"2d9d00a5-0e05-495c-8914-38ce7a0939e5","cardType":"basic","points":2,"front":"Why do quadratic simultaneous equations typically result in two solutions?","back":"Because a straight line can intersect a quadratic curve (like a circle) at two distinct points.","mcqOptions":[{"text":"Because a straight line can intersect a quadratic curve (like a circle) at two distinct points.","correct":true,"explanation":"Each intersection satisfies both equations. A line can cross a quadratic curve in two places, producing two distinct coordinate pairs that solve the simultaneous equations."},{"text":"Because every quadratic equation must have two distinct roots.","correct":false,"explanation":"A quadratic can instead have one repeated real root or no real roots, depending on its discriminant. Two solutions typically arise when the line intersects the curve twice."},{"text":"Because each coordinate pair can be written in two different orders.","correct":false,"explanation":"Coordinates cannot be reversed freely because $(x,y)$ and $(y,x)$ usually represent different points. Two solutions arise from two distinct intersections of the line and curve."},{"text":"Because a straight line always crosses the coordinate axes at two distinct points.","correct":false,"explanation":"Axis intercepts are unrelated to whether the line meets the quadratic curve. The simultaneous equations have two solutions when their graphs intersect at two distinct points."}]},{"id":"3fa5595d-6833-4d4b-825a-077bb8c3ebde","cardType":"basic","points":1,"front":"What is the first step in solving a quadratic simultaneous equation algebraically?","back":"Rearrange the linear equation to make one variable the subject.","mcqOptions":[{"text":"Rearrange the linear equation to make one variable the subject.","correct":true,"explanation":"Isolating one variable in the linear equation creates an expression that can be substituted into the quadratic equation, leaving a single equation in one unknown."},{"text":"Rearrange the quadratic equation to make both variables the subject.","correct":false,"explanation":"A quadratic equation usually cannot make both variables the subject in a useful single expression. Instead, rearrange the simpler linear equation to isolate one variable for substitution."},{"text":"Factorise the linear equation to find both variables immediately.","correct":false,"explanation":"A linear equation is not normally solved by factorisation and cannot determine both variables alone. Rearrange it to make one variable the subject before substituting."},{"text":"Add the two equations to eliminate every squared term.","correct":false,"explanation":"Adding the equations does not generally eliminate the squared term because only the quadratic equation contains it. First isolate a variable in the linear equation and substitute."}]},{"id":"687f9306-d7ea-421d-9378-a2f6ac85615f","cardType":"cloze","points":null,"front":"Substitution turns two equations into one equation with [ ].","back":"Substitution turns two equations into one equation with one unknown.","mcqOptions":[{"text":"one unknown","correct":true,"explanation":"Replacing one variable using the rearranged linear equation removes that variable from the quadratic equation, producing a single equation containing only one unknown."},{"text":"two unknowns","correct":false,"explanation":"Leaving two unknowns means the substitution has not fully replaced one variable. Correct substitution produces one equation in one unknown, which can then be solved."},{"text":"three unknowns","correct":false,"explanation":"Substitution does not introduce an additional variable; it replaces one variable with an equivalent expression. The resulting equation therefore contains one unknown, not three."},{"text":"no unknowns","correct":false,"explanation":"Substitution removes one of the two variables, but the remaining variable must still be found. It therefore creates an equation with one unknown rather than none."}]},{"id":"342b3fe0-6fe8-47f1-824b-620652e7c722","cardType":"basic","points":2,"front":"How are the matching $y$-values found once the $x$-values are solved?","back":"Substitute each $x$-value into the rearranged linear equation.","mcqOptions":[{"text":"Substitute each $x$-value into the rearranged linear equation.","correct":true,"explanation":"The rearranged linear equation expresses $y$ directly in terms of $x$. Substituting each solved $x$-value therefore gives its corresponding $y$-value unambiguously."},{"text":"Substitute each $x$-value into the original quadratic equation.","correct":false,"explanation":"Using the quadratic equation may produce multiple possible $y$-values, including values that do not satisfy the line. The rearranged linear equation gives the matching value directly."},{"text":"Substitute each $x$-value into the quadratic formula.","correct":false,"explanation":"The quadratic formula is used to solve a quadratic for its unknown, not to recover the corresponding second coordinate. Use the rearranged linear equation to find each $y$-value."},{"text":"Substitute each $x$-value into the discriminant expression.","correct":false,"explanation":"The discriminant indicates the number and type of quadratic roots but does not calculate matching coordinates. Substitute each root into the rearranged linear equation instead."}]},{"id":"453ab1b8-1ee5-421e-9b92-5d9413d12322","cardType":"basic","points":1,"front":"In what format should final solutions to simultaneous equations be given?","back":"As coordinate pairs $(x, y)$.","mcqOptions":[{"text":"As coordinate pairs $(x, y)$.","correct":true,"explanation":"Each solution consists of an $x$-value and its corresponding $y$-value, so writing $(x,y)$ keeps the values correctly matched as a point satisfying both equations."},{"text":"As separate lists $x, y$.","correct":false,"explanation":"Separate lists can lose the matching between each $x$-value and its corresponding $y$-value. Each complete solution should be written as an ordered coordinate pair $(x,y)$."},{"text":"As unordered pairs $(x, y)$.","correct":false,"explanation":"The order matters because the first coordinate is $x$ and the second is $y$. Solutions must therefore be given as ordered coordinate pairs, not unordered pairs."},{"text":"As single values $(x+y)$.","correct":false,"explanation":"Combining the coordinates into one sum loses the individual values and does not identify a point. Each solution must retain both values as a coordinate pair $(x,y)$."}]},{"id":"b05d70dd-90ec-4216-8615-896dd04274b9","cardType":"basic","points":2,"front":"Explain the common mistake involving square roots and the equation $x^2 + y^2 = 10$.","back":"Incorrectly assuming $x + y = \\sqrt{10}$; square roots do not distribute across addition.","mcqOptions":[{"text":"Incorrectly assuming $x + y = \\sqrt{10}$; square roots do not distribute across addition.","correct":true,"explanation":"In general, $\\sqrt{x^2+y^2}$ is not equal to $x+y$. The entire sum is square-rooted, so the original equation cannot be simplified in this way."},{"text":"Incorrectly assuming $x^2 + y^2 = 100$; square roots must be squared across addition.","correct":false,"explanation":"Squaring the right-hand side without applying an equivalent operation to the left changes the equation. The common error is instead treating $\\sqrt{x^2+y^2}$ as $x+y$."},{"text":"Incorrectly assuming $x + y = 10$; exponents do not distribute across addition.","correct":false,"explanation":"Simply removing the squares is invalid, but it is not the stated square-root misconception. The common error gives $x+y=\\sqrt{10}$ by distributing the square root incorrectly."},{"text":"Incorrectly assuming $xy = \\sqrt{10}$; square roots do not convert addition into multiplication.","correct":false,"explanation":"Taking a square root does not turn a sum into a product. The usual misconception is that it distributes across the sum, leading incorrectly to $x+y=\\sqrt{10}$."}]},{"id":"5fadbf66-56e8-432b-9693-43f31f0b4da8","cardType":"basic","points":1,"front":"What does the quadratic equation $x^2 + y^2 = r^2$ represent graphically?","back":"A circle centred at the origin $(0,0)$.","mcqOptions":[{"text":"A circle centred at the origin $(0,0)$.","correct":true,"explanation":"All points satisfying $x^2+y^2=r^2$ are a distance $r$ from $(0,0)$ by Pythagoras, which is exactly the definition of a circle."},{"text":"An ellipse centred at the origin $(0,0)$.","correct":false,"explanation":"An ellipse generally has differently scaled $x^2$ and $y^2$ terms. Equal coefficients produce points at the same radius $r$ from the origin, forming a circle."},{"text":"A parabola with its vertex at the origin $(0,0)$.","correct":false,"explanation":"A parabola typically has only one variable squared, whereas both variables are squared and added here. This equation describes a circle centred at the origin."},{"text":"A circle centred at the point $(r,r)$.","correct":false,"explanation":"A circle centred at $(r,r)$ would involve $(x-r)^2+(y-r)^2$. With no horizontal or vertical shifts, the centre is the origin $(0,0)$."}]},{"id":"404453db-862b-48ac-9488-87654cee244e","cardType":"basic","points":1,"front":"If a question requires answers to 3 significant figures, what does this imply about the quadratic?","back":"The quadratic will not factorise, and you must use the quadratic formula.","mcqOptions":[{"text":"The quadratic will not factorise, and you must use the quadratic formula.","correct":true,"explanation":"A requirement for 3 significant figures indicates that the roots are not expected to be exact factorisable values, so the quadratic formula is needed to calculate decimal solutions."},{"text":"The quadratic will factorise, and you must solve it by factorising.","correct":false,"explanation":"Factorising normally produces exact roots, so answers would not need to be requested to 3 significant figures. The quadratic will not factorise, requiring the quadratic formula instead."},{"text":"The quadratic has a repeated root, and you must use completing the square.","correct":false,"explanation":"A repeated root is a special case with a discriminant of zero, not something implied by decimal rounding. The expected implication is non-factorisation and use of the quadratic formula."},{"text":"The quadratic has integer roots, and you must use inspection.","correct":false,"explanation":"Integer roots would usually allow straightforward factorisation and exact answers. A request for 3 significant figures instead suggests non-factorisable roots calculated using the quadratic formula."}]},{"id":"85f7dd5b-6322-416e-a231-3c1eb4d5418c","cardType":"cloze","points":null,"front":"In the number $-0.120$, the final zero is [ ].","back":"In the number $-0.120$, the final zero is significant.","mcqOptions":[{"text":"significant","correct":true,"explanation":"A trailing zero to the right of a decimal follows non-zero digits and shows the stated precision. It therefore counts as a significant figure."},{"text":"not significant","correct":false,"explanation":"Trailing zeros after a decimal point are counted because they communicate precision. The final zero in this decimal is significant, giving the number three significant figures."},{"text":"a placeholder","correct":false,"explanation":"Placeholder zeros occur before the first non-zero digit, such as the zeros in 0.0012. Here, the final zero follows non-zero digits and is significant."},{"text":"ignored","correct":false,"explanation":"Only leading zeros before the first non-zero digit are ignored when counting significant figures. The final zero follows the digits 1 and 2, so it is significant."}]},{"id":"7ee27486-e38b-42fe-92de-f2ee937dda46","cardType":"basic","points":2,"front":"In the expansion of $(5 - 4x)^2$, what are the resulting three terms?","back":"$$25 - 40x + 16x^2$","mcqOptions":[{"text":"$$25 - 40x + 16x^2$","correct":true,"explanation":"Using $(a-b)^2=a^2-2ab+b^2$ gives $5^2-2(5)(4x)+(4x)^2$, which simplifies to $25-40x+16x^2$."},{"text":"$$25 - 20x + 16x^2$","correct":false,"explanation":"The middle term must include the factor of 2 from $(a-b)^2=a^2-2ab+b^2$. Therefore it is $-40x$, and the correct expansion is $25-40x+16x^2$."},{"text":"$$25 + 40x + 16x^2$","correct":false,"explanation":"The subtraction inside the bracket makes the cross-term negative: $2(5)(-4x)=-40x$. The correct expansion is therefore $25-40x+16x^2$."},{"text":"$$25 - 40x - 16x^2$","correct":false,"explanation":"Squaring $-4x$ gives positive $16x^2$, not a negative term. Including the negative cross-term, the correct expansion is $25-40x+16x^2$."}]},{"id":"80d29ab3-05f3-4108-9521-f1f7cbb5e36c","cardType":"cloze","points":null,"front":"In the expression $2x^2 - (1 - 6x + 9x^2)$, the negative sign must be applied to [ ] inside the bracket.","back":"In the expression $2x^2 - (1 - 6x + 9x^2)$, the negative sign must be applied to every term inside the bracket.","mcqOptions":[{"text":"every term","correct":true,"explanation":"Subtracting a bracket is equivalent to multiplying the entire bracket by $-1$. Consequently, each term changes sign when the bracket is removed."},{"text":"only the first term","correct":false,"explanation":"The minus sign represents multiplication of the whole bracket by $-1$, not just its first entry. It must therefore be applied to every term inside the bracket."},{"text":"only the middle term","correct":false,"explanation":"Applying the outside minus to only the middle term leaves the constant and quadratic signs unchanged incorrectly. Every term in the bracket must have its sign reversed."},{"text":"only the final term","correct":false,"explanation":"The minus sign before a bracket distributes across the entire expression, rather than only the last entry. Every term inside the bracket must change sign."}]},{"id":"a80a18e1-50d9-4905-99fc-290978322d6b","cardType":"cloze","points":null,"front":"A quadratic equation must be equal to [ ] before using the quadratic formula or factorising.","back":"A quadratic equation must be equal to 0 before using the quadratic formula or factorising.","mcqOptions":[{"text":"0","correct":true,"explanation":"The quadratic formula and standard factorisation apply to an equation written as $ax^2+bx+c=0$. All terms must therefore be collected on one side first."},{"text":"1","correct":false,"explanation":"Setting a quadratic equal to 1 does not place it in the required standard form. Rearrange all terms so that one side equals 0 before solving."},{"text":"x","correct":false,"explanation":"Leaving $x$ on the other side means the equation has not yet been arranged in standard quadratic form. Move it across so that the equation equals 0."},{"text":"-1","correct":false,"explanation":"The required standard form is $ax^2+bx+c=0$, not an equation equal to a chosen negative constant. Rearrange the constant so that one side equals 0."}]},{"id":"83eaa51c-cdbd-490a-affc-dbd35d089750","cardType":"cloze","points":null,"front":"If the value under the square root in the quadratic formula is negative, the line and curve [ ].","back":"If the value under the square root in the quadratic formula is negative, the line and curve do not intersect.","mcqOptions":[{"text":"do not intersect","correct":true,"explanation":"A negative discriminant gives no real solutions for the intersection equation. Graphically, this means there are no real coordinate points shared by the line and curve."},{"text":"intersect at two points","correct":false,"explanation":"Two intersections correspond to two distinct real roots and require a positive discriminant. A negative discriminant gives no real roots, so the line and curve do not intersect."},{"text":"touch at exactly one point","correct":false,"explanation":"Touching once represents a repeated real root, which occurs when the discriminant equals zero. A negative discriminant instead means the line and curve do not intersect."},{"text":"intersect at the origin","correct":false,"explanation":"Passing through the origin would require $(0,0)$ to satisfy both equations and is unrelated to a negative discriminant. A negative value means there are no intersections."}]},{"id":"7323b193-fb3a-4d70-8f48-a01f5c3a38af","cardType":"basic","points":2,"front":"What is indicated graphically if the quadratic equation results in a repeated root?","back":"The line is a tangent to the curve (they touch at exactly one point).","mcqOptions":[{"text":"The line is a tangent to the curve (they touch at exactly one point).","correct":true,"explanation":"A repeated root gives one shared $x$-coordinate counted twice. Graphically, the line meets the curve at exactly one point without crossing it, making the line tangent."},{"text":"The line is a secant to the curve (they cross at exactly two points).","correct":false,"explanation":"A secant produces two distinct intersection points and therefore two distinct real roots. A repeated root gives only one point of contact, so the line is tangent."},{"text":"The line is parallel to the curve (they never meet).","correct":false,"explanation":"No meeting points would mean there are no real roots, rather than a repeated root. A repeated root means one point of contact, making the line tangent."},{"text":"The line is a normal to the curve (they meet at a right angle).","correct":false,"explanation":"A normal concerns a line perpendicular to a curve’s tangent and is not implied by a repeated root. One repeated intersection means the given line is tangent."}]},{"id":"50e6cdca-ef67-4f0f-9d92-7a08de49149b","cardType":"cloze","points":null,"front":"To solve [ ] means to use methods like substitution and rearranging rather than using a graph. house.","back":"To solve algebraically means to use methods like substitution and rearranging rather than using a graph. house.","mcqOptions":[{"text":"algebraically","correct":true,"explanation":"Substitution, elimination and rearrangement manipulate equations using mathematical symbols and operations. These are algebraic methods, as opposed to reading intersections from a graph."},{"text":"graphically","correct":false,"explanation":"A graphical solution is obtained by plotting both relations and reading their intersection points. Substitution and rearrangement instead mean solving algebraically."},{"text":"numerically","correct":false,"explanation":"Numerical methods approximate solutions through repeated calculations or trials. Substitution and exact rearrangement are algebraic procedures, so the required term is algebraically."},{"text":"iteratively","correct":false,"explanation":"Iterative solving repeatedly applies a formula to approach an approximate solution. Direct substitution and rearrangement are instead methods of solving algebraically."}]},{"id":"f5973d63-b022-4768-be51-b0e9ee51ac6b","cardType":"cloze","points":null,"front":"The denominator of the quadratic formula is [ ].","back":"The denominator of the quadratic formula is $2a$.","mcqOptions":[{"text":"$$2a$","correct":true,"explanation":"In $ax^2 + bx + c = 0$, the quadratic formula divides the entire numerator $-b \\pm \\sqrt{b^2 - 4ac}$ by $2a$."},{"text":"$$2b$","correct":false,"explanation":"This confuses the linear coefficient with the quadratic coefficient. The denominator is $2a$, where $a$ is the coefficient of $x^2$."},{"text":"$$a$","correct":false,"explanation":"This omits the factor of $2$ in the quadratic formula. The whole numerator must be divided by $2a$, not merely by $a$."},{"text":"$$4a$","correct":false,"explanation":"This incorrectly doubles the denominator’s numerical factor. The denominator of the quadratic formula is $2a$, while the discriminant contains the factor $4ac$."}]},{"id":"84bd08a9-9e79-457f-98a6-fb7bb0af39a2","cardType":"cloze","points":null,"front":"The part of the quadratic formula inside the square root is [ ].","back":"The part of the quadratic formula inside the square root is $b^2 - 4ac$.","mcqOptions":[{"text":"$$b^2 - 4ac$","correct":true,"explanation":"The expression $b^2 - 4ac$ is the discriminant. It appears beneath the square root in the quadratic formula and determines the number of real roots."},{"text":"$$b^2 + 4ac$","correct":false,"explanation":"The sign between the terms has been changed incorrectly. The discriminant beneath the square root is $b^2 - 4ac$, with subtraction rather than addition."},{"text":"$$b - 4ac$","correct":false,"explanation":"This fails to square the coefficient $b$. The expression inside the square root is $b^2 - 4ac$, not $b - 4ac$."},{"text":"$$4ac - b^2$","correct":false,"explanation":"This reverses the order of the subtraction and therefore changes the discriminant’s sign. The quadratic formula uses $b^2 - 4ac$."}]},{"id":"819b789a-607c-42bb-bce2-0d01019404d4","cardType":"basic","points":1,"front":"Rearrange $x - y = 3$ to make $y$ the subject.","back":"$$y = x - 3$","mcqOptions":[{"text":"$$y = x - 3$","correct":true,"explanation":"Adding $y$ to both sides and subtracting $3$ from both sides isolates $y$, giving $y = x - 3$."},{"text":"$$y = x + 3$","correct":false,"explanation":"The constant’s sign has been changed incorrectly. From $x - y = 3$, isolating $y$ gives $y = x - 3$, not an addition of $3$."},{"text":"$$y = 3 - x$","correct":false,"explanation":"This results from moving terms without changing both signs consistently. Multiplying $-y = 3 - x$ by $-1$ gives $y = x - 3$."},{"text":"$$y = -x - 3$","correct":false,"explanation":"Both terms have been given incorrect signs. Rearranging produces $-y = 3 - x$, so multiplying throughout by $-1$ gives $y = x - 3$."}]},{"id":"dc83305a-4836-4574-ad14-94e62bc0922c","cardType":"basic","points":1,"front":"Rearrange $2y = 1 - 3x$ to make $y$ the subject.","back":"$$y = \\frac{1 - 3x}{2}$","mcqOptions":[{"text":"$$y = \\frac{1 - 3x}{2}$","correct":true,"explanation":"Dividing every term in $2y = 1 - 3x$ by $2$ isolates $y$, giving $y = \\frac{1 - 3x}{2}$."},{"text":"$$y = 1 - 3x$","correct":false,"explanation":"This forgets to divide the right-hand side by $2$. Since the left side is $2y$, both terms on the right must be halved."},{"text":"$$y = \\frac{1}{2} - 3x$","correct":false,"explanation":"Only the constant has been divided by $2$, leaving the $x$ term unchanged. Every term must be divided, giving $y = \\frac{1 - 3x}{2}$."},{"text":"$$y = \\frac{1 + 3x}{2}$","correct":false,"explanation":"The sign of the $3x$ term has been changed unnecessarily. Division by $2$ does not alter its sign, so the numerator remains $1 - 3x$."}]}]}],"examCountdown":null,"heading":"Quadratic Simultaneous Equations","title":"Quadratic Simultaneous Equations Flashcards","topbar":{"backHref":"/igcse/maths/cie/flashcards"},"topicSidebar":"$L38"}]}],"topicId":"9b1881f2-73db-4b0c-bee3-2cb3125ae984","walled":"$L39"}],"$L3a","$L3b"]